# How can I measure the mode content of a LASER beam?

The symmetry of a stable optical resonator determine that Hermite–Gaussian functions will be formed. In multimode optical fibres, these will be LP modes instead. I am interested wether one can measure the actual mode content, e.g. determine which mode contributes which fraction of the intensity (or electric field + phase). Is this experimentally possible? How fast does the content change over time (between the different modes)? Can one measure this without prior knowledge of the mode shape?

You can't infer the mode content, at least practically and reliably, from intensity measurements alone even if you do know the modeshapes unless the number of modes present is small. You can, for example, estimate the purity of a single, fundamental mode if the degree of contamination is small. For a small number of modes, there is a trick that you can use to help you, especially in the case of a fiber.

Suppose the modeshapes as a function of transverse position $\vec{r}$ are $\psi_i(\vec{r})$. Then, assuming you don't want to know phase, you wish to find the $|\alpha_i|^2$ given a measurement of:

$$|\psi^t(\vec{r}_j)|^2 = \left|\sum\limits_i \alpha_i\,\psi_i(\vec{r}_j)\right|^2\tag{1}$$

We expand the above to find:

$$|\psi^t(\vec{r}_j)|^2(t) = \sum\limits_i |\alpha_i|^2\,|\psi_i(\vec{r}_j)|^2 + \sum\limits_{j\neq i} 2 \mathrm{Re}\left(\alpha_i\,\alpha_j^\ast\,e^{i\,\phi_{i\,j}(t)}\right)\tag{2}$$

where the $\phi_{i\,j}(t)$ are randomly time-varying phase delays between modes that arise as the fiber strains or as the laser cavity mode-hops owing to temperature and other environmental variations. You can speed this variation up by deliberately vibrating the laser cavity or piezoelectrically modulating the fiber, measuring the total intensity over a transverse plane and averaging the measured values of $|\psi^t(\vec{r}_j)|^2$ as you do so. The idea here is that the off-diagonal terms (the terms in the $\sum\limits_{j\neq i}$ that involve the $e^{i\,\phi_{i\,j}(t)}$ factors) will average to nought owing to the random variations in the $\phi_{i\,j}(t)$ with time. Once you have measured this average (you know when you have because the average stops varying with time), you have:

$$\langle|\psi^t(\vec{r}_j)|^2\rangle = \sum\limits_i |\alpha_i|^2\,|\psi_i(\vec{r}_j)|^2\tag{3}$$

which, given enough points $j$ on the transverse plane, and given we know the mode intensity shapes $|\psi_i(\vec{r}_j)|^2$, can be analyzed by a linear regression for the weights $|\alpha_i|^2$ of the mode intensities $|\psi_i(\vec{r}_j)|^2$. So you can in theory infer the $|\alpha_i|^2$ in this way. The practical problem is that, unlike the mode amplitudes, the mode intensities are often not even close to being orthogonal, so that the linear regression, unless there are only very few modes of very different intensity profile present, will become badly numerically conditioned. You should use a singular value decomposition for your linear regression to check this.

As for how fast the content changes: for a fiber, the mode intensity content should not change unless the illumination conditions change. The phases between the modes will change at a speed in proportion to the fiber length: for a very long fiber of several kilometers, the phases between the modes vary over many, many cycles per second, so the cross terms in (2) can vary in at hundreds or even kilohertz rates. For a laser, the answer to this question depends very much on the physics giving rise to the variation. Slow, temperature induced changes are typically at a few hertz rate. Mode hopping can be very violent, and even happen in the megahertz range.

Follow Up Question by OP

.... How would it change if I were able to measure the phase as well? Is it then possible?

Essentially you then have access to $\psi(\vec{x})$ as a complex field and you're doing interferometry. If you know the modes, they are orthogonal, and you can then find their complex weights by linear regresssion, which will now work well owing to the orthogonality. But you do need to know the mode shapes as you can see from the following: a complex function can be resolved into a superposition of different complete orthonormal sets: the orthonormal sets are not unique. Do a nontrivial unitary transformation on any basis and you have an equally workable - and different - basis.

• Thanks for this detailed answer. How would it change if I were able to measure the phase as well? Is it then possible? – DaP Nov 30 '16 at 7:18
• @DaP See my updates. – Selene Routley Dec 1 '16 at 23:10